It has been known for some time in the mathematical literature that quadrangulated surfaces can be encoded in terms of labeled trees. Building on this correspondence I will present a simple model of random trees that includes in a natural way both (Euclidean) DT and CDT in two dimensions. A double scaling limit leads to continuum CDT with spatial topology change, providing an explicit realization of so-called generalized CDT. To emphasize the close friendship between CDT and trees I will show how an interesting identity for generalized CDT, which some time ago was painstakingly derived in the CDT setting, turns into a triviality when reinterpreted in terms of trees (or rather: planar maps).

1. In Search for Quantum Gravity: CDT & Friends,
Dec. 11-14, 2012, Nijmegen
CDT & Trees
Timothy Budd
Niels Bohr Institute, Copenhagen.
budd@nbi.dk
Collaboration with J. Ambjørn

2. Outline
Quadrangulations & Trees: ←→
0
000
0
0 0
00
CDT & Trees: ←→
Generalized CDT & Trees: ←→
0
000
0
0 0
00
Loop amplitudes & Planar maps: ←→
0
0

3. The Cori–Vauquelin–Schaeffer bijection
[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
Quadrangulation

4. The Cori–Vauquelin–Schaeffer bijection
[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
Quadrangulation → mark a point

5. The Cori–Vauquelin–Schaeffer bijection
[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t 1
t 1
t t
t 1
t 1
Quadrangulation → mark a point → distance labeling

6. The Cori–Vauquelin–Schaeffer bijection
[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t 1
t 1
t t
t 1
t 1
Quadrangulation → mark a point → distance labeling → apply rules

7. The Cori–Vauquelin–Schaeffer bijection
[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t 1
t 1
t t
t 1
t 1
Quadrangulation → mark a point → distance labeling → apply rules

8. The Cori–Vauquelin–Schaeffer bijection
[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t 1
t 1
t t
t 1
t 1
Quadrangulation → mark a point → distance labeling → apply rules

9. The Cori–Vauquelin–Schaeffer bijection
[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t 1
t 1
t t
t 1
t 1
Quadrangulation → mark a point → distance labeling → apply rules
→ labelled tree.

10. The Cori–Vauquelin–Schaeffer bijection
[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t 1
t 1
t t
t 1
t 1
Quadrangulation → mark a point → distance labeling → apply rules
→ labelled tree.

11. The Cori–Vauquelin–Schaeffer bijection
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[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t 1
t 1
t t
t 1
t 1
Quadrangulation → mark a point → distance labeling → apply rules
→ labelled tree.
Labelled tree

12. The Cori–Vauquelin–Schaeffer bijection
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[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t 1
t 1
t t
t 1
t 1
Quadrangulation → mark a point → distance labeling → apply rules
→ labelled tree.
Labelled tree → add squares

13. The Cori–Vauquelin–Schaeffer bijection
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[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t 1
t 1
t t
t 1
t 1
Quadrangulation → mark a point → distance labeling → apply rules
→ labelled tree.
Labelled tree → add squares → identify corners

14. The Cori–Vauquelin–Schaeffer bijection
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[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t 1
t 1
t t
t 1
t 1
Quadrangulation → mark a point → distance labeling → apply rules
→ labelled tree.
Labelled tree → add squares → identify corners

15. The Cori–Vauquelin–Schaeffer bijection
[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t 1
t 1
t t
t 1
t 1
Quadrangulation → mark a point → distance labeling → apply rules
→ labelled tree.
Labelled tree → add squares → identify corners → quadrangulation.

16. Rooting the tree (Miermont, Bouttier, Guitter, Le Gall,...)

17. Rooting the tree (Miermont, Bouttier, Guitter, Le Gall,...)
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19. Rooting the tree (Miermont, Bouttier, Guitter, Le Gall,...)
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We will be using the bijection:
Quadrangulations with origin
and marked edge
↔
Rooted planar trees
labelled by +, 0, −
×Z2

21. Causal triangulations/quadrangulations
[Krikun, Yambartsev ’08]
[Durhuus, Jonsson, Wheater ’09]
(Wednesday’s talks!)

22. Causal triangulations/quadrangulations
[Krikun, Yambartsev ’08]
[Durhuus, Jonsson, Wheater ’09]
(Wednesday’s talks!)

23. Causal triangulations/quadrangulations
1
2
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11
12
13
t
[Krikun, Yambartsev ’08]
[Durhuus, Jonsson, Wheater ’09]
(Wednesday’s talks!)
If we take the origin at t = 0 and the root at final time, all edges of
the tree are labeled −.

24. Causal triangulations/quadrangulations
1
2
3
4
5
6
7
8
9
10
11
12
13
t
[Krikun, Yambartsev ’08]
[Durhuus, Jonsson, Wheater ’09]
(Wednesday’s talks!)
If we take the origin at t = 0 and the root at final time, all edges of
the tree are labeled −.
Quadrangulations ↔ labeled trees. Causal quadr. ↔ trees.

25. Causal triangulations/quadrangulations
1
2
3
4
5
6
7
8
9
10
11
12
13
t
[Krikun, Yambartsev ’08]
[Durhuus, Jonsson, Wheater ’09]
(Wednesday’s talks!)
If we take the origin at t = 0 and the root at final time, all edges of
the tree are labeled −.
Quadrangulations ↔ labeled trees. Causal quadr. ↔ trees.
As a direct consequence: With N squares we can build
#
N
= C(N), #
N
= 2C(N)3N
, C(N) =
1
N + 1
2N
N

26. Including boundaries [Bettinelli ’11]
A quadrangulations with
boundary length 2l

27. Including boundaries [Bettinelli ’11]
A quadrangulations with
boundary length 2l and an
origin.

28. Including boundaries [Bettinelli ’11]
A quadrangulations with
boundary length 2l and an
origin.
Applying the same prescription
we obtain a forest rooted at the
boundary.
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89
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29. Including boundaries [Bettinelli ’11]
A quadrangulations with
boundary length 2l and an
origin.
Applying the same prescription
we obtain a forest rooted at the
boundary.
The labels on the boundary
arise from a (closed) random
walk.
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30. Including boundaries [Bettinelli ’11]
A quadrangulations with
boundary length 2l and an
origin.
Applying the same prescription
we obtain a forest rooted at the
boundary.
The labels on the boundary
arise from a (closed) random
walk.
A (possibly empty) tree grows
at the end of every +-edge.
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5 4
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9
89
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6

31. Including boundaries [Bettinelli ’11]
A quadrangulations with
boundary length 2l and an
origin.
Applying the same prescription
we obtain a forest rooted at the
boundary.
The labels on the boundary
arise from a (closed) random
walk.
A (possibly empty) tree grows
at the end of every +-edge.
There is a bijection [Bettinelli]
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5 4
3
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4
3
2
3
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3
4
5
6
7
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9
89
8
7
6
Quadrangulations with origin
and boundary length 2l
↔ {(+, −)-sequences} × {tree}
l

32. Disk amplitudes

33. Disk amplitudes

34. Disk amplitudes
w(g, l) = z(g)l
w(g, x) =
∞
l=0
w(g, l)xl
=
1
1 − z(g)x
w(g, l) =
2l
l
z(g)l
w(g, x) =
1
1 − 4z(g)x

35. Disk amplitudes
w(g, l) = z(g)l
w(g, x) =
∞
l=0
w(g, l)xl
=
1
1 − z(g)x
w(g, l) =
2l
l
z(g)l
w(g, x) =
1
1 − 4z(g)x
Generating function
for unlabeled trees:
z(g) =
1 −
√
1 − 4g
2g
0
000
0
0 0
00
Generating function
for labeled trees:
z(g) =
1 −
√
1 − 12g
6g

36. Continuum limit
w(g, x) =
1
1 − zx
w(g, x) =
1
√
1 − 4zx
z(g) = 1−
√
1−4g
2g
0
000
0
0 0
00
z(g) = 1−
√
1−12g
6g

37. Continuum limit
w(g, x) =
1
1 − zx
w(g, x) =
1
√
1 − 4zx
z(g) = 1−
√
1−4g
2g
0
000
0
0 0
00
z(g) = 1−
√
1−12g
6g
Expanding around critical point in terms of “lattice spacing” :
g = gc (1 − Λ 2
), z(g) = zc (1 − Z ), x = xc (1 − X )

38. Continuum limit
w(g, x) =
1
1 − zx
WΛ(X) =
1
X + Z
w(g, x) =
1
√
1 − 4zx
WΛ(X) =
1
√
X + Z
z(g) = 1−
√
1−4g
2g
Z =
√
Λ
0
000
0
0 0
00
z(g) = 1−
√
1−12g
6g
Z =
√
Λ
Expanding around critical point in terms of “lattice spacing” :
g = gc (1 − Λ 2
), z(g) = zc (1 − Z ), x = xc (1 − X )

39. Continuum limit
w(g, x) =
1
1 − zx
WΛ(X) =
1
X + Z
w(g, x) =
1
√
1 − 4zx
WΛ(X) =
1
√
X + Z
z(g) = 1−
√
1−4g
2g
Z =
√
Λ
0
000
0
0 0
00
z(g) = 1−
√
1−12g
6g
Z =
√
Λ
Expanding around critical point in terms of “lattice spacing” :
g = gc (1 − Λ 2
), z(g) = zc (1 − Z ), x = xc (1 − X )
CDT disk amplitude: WΛ(X) = 1
X+
√
Λ

40. Continuum limit
w(g, x) =
1
1 − zx
WΛ(X) =
1
X + Z
w(g, x) =
1
√
1 − 4zx
WΛ(X) =
1
√
X + Z
z(g) = 1−
√
1−4g
2g
Z =
√
Λ
0
000
0
0 0
00
z(g) = 1−
√
1−12g
6g
Z =
√
Λ
Expanding around critical point in terms of “lattice spacing” :
g = gc (1 − Λ 2
), z(g) = zc (1 − Z ), x = xc (1 − X )
CDT disk amplitude: WΛ(X) = 1
X+
√
Λ
DT disk amplitude with marked point: WΛ(X) = 1√
X+
√
Λ
. Integrate
w.r.t. Λ to remove mark: WΛ(X) = 2
3 (X − 1
2
√
Λ) X +
√
Λ.

41. Generalized CDT [Ambjørn, Loll, Westra, Zohren ’07]
Assign coupling a to the local maxima of the
distance function.

42. Generalized CDT [Ambjørn, Loll, Westra, Zohren ’07]
Assign coupling a to the local maxima of the
distance function.
In terms of labeled trees:
5 4
44
4
4 4
5
3
3
5
5

43. Generalized CDT [Ambjørn, Loll, Westra, Zohren ’07]
Assign coupling a to the local maxima of the
distance function.
In terms of labeled trees:
5 4
44
4
4 4
5
3
3
5
5

44. Generalized CDT [Ambjørn, Loll, Westra, Zohren ’07]
Assign coupling a to the local maxima of the
distance function.
In terms of labeled trees:
5 4
4
5
5
5

45. Generalized CDT [Ambjørn, Loll, Westra, Zohren ’07]
Assign coupling a to the local maxima of the
distance function.
In terms of labeled trees:
Local maximum if +/0
coming in and any number
of 0/−’s going out.
0
0
0
0
0

46. Generalized CDT [Ambjørn, Loll, Westra, Zohren ’07]
Assign coupling a to the local maxima of the
distance function.
In terms of labeled trees:
Local maximum if +/0
coming in and any number
of 0/−’s going out.
0
0
0
0
0
We can write down equations for the generating functions
= g
∞
k=0
+ 0
+
k
= g 1 − − 0
−
−1

47. Generalized CDT [Ambjørn, Loll, Westra, Zohren ’07]
Assign coupling a to the local maxima of the
distance function.
In terms of labeled trees:
Local maximum if +/0
coming in and any number
of 0/−’s going out.
0
0
0
0
0
We can write down equations for the generating functions
= g
∞
k=0
+ 0
+
k
= g 1 − − 0
−
−1
= 0
= g
∞
k=0
+ 0
+
k
+ (a − 1)
∞
k=0
+ 0
k
= + g(a − 1) 1 − − 0
−1

48. = g 1 − − 0
−
−1
= 0
= + g(a − 1) 1 − − 0
−1
Combine into one equation for z−(g, a) = :
3z4
− − 4z3
− + (1 + 2g(1 − 2a))z2
− − g2
= 0

49. = g 1 − − 0
−
−1
= 0
= + g(a − 1) 1 − − 0
−1
Combine into one equation for z−(g, a) = :
3z4
− − 4z3
− + (1 + 2g(1 − 2a))z2
− − g2
= 0
Phase diagram for weighted labeled trees (constant a):
a 0
a 1
CDT
DT
0
1
12
1
8
1
6
1
4
1
24
5
24
g
1
6
1
2
1
3
z g, a

50. = g 1 − − 0
−
−1
= 0
= + g(a − 1) 1 − − 0
−1
Combine into one equation for z−(g, a) = :
3z4
− − 4z3
− + (1 + 2g(1 − 2a))z2
− − g2
= 0
Phase diagram for weighted labeled trees (constant a):
2 z3
3 z4
g2
0
a 0
a 1
0
1
12
1
8
1
6
1
4
1
24
5
24
g
1
6
1
2
1
3
z g, a

51. = g 1 − − 0
−
−1
= 0
= + g(a − 1) 1 − − 0
−1
Combine into one equation for z−(g, a) = :
3z4
− − 4z3
− + (1 + 2g(1 − 2a))z2
− − g2
= 0
Phase diagram for weighted labeled trees (constant a):
2 z3
3 z4
g2
0
a 0
a 1
0
1
12
1
8
1
6
1
4
1
24
5
24
g
1
6
1
2
1
3
z g, a

52. N = 2000, a = 0, Nmax = 0

53. N = 5000, a = 0.00007, Nmax = 11

54. N = 7000, a = 0.0002, Nmax = 37

55. N = 7000, a = 0.004, Nmax = 221

56. N = 4000, a = 0.02, Nmax = 362

57. N = 2500, a = 1, Nmax = 1216

58. The number of local maxima Nmax (a) scales with N at the critical
point,
Nmax (a) N
N
= 2
a
2
2/3
+ O(a),
Nmax (a = 1) N
N
= 1/2

59. The number of local maxima Nmax (a) scales with N at the critical
point,
Nmax (a) N
N
= 2
a
2
2/3
+ O(a),
Nmax (a = 1) N
N
= 1/2
Therefore, to obtain a finite continuum density of critical points one
should scale a ∝ N−3/2
, i.e. a = gs
3
as observed in [ALWZ ’07].

60. The number of local maxima Nmax (a) scales with N at the critical
point,
Nmax (a) N
N
= 2
a
2
2/3
+ O(a),
Nmax (a = 1) N
N
= 1/2
Therefore, to obtain a finite continuum density of critical points one
should scale a ∝ N−3/2
, i.e. a = gs
3
as observed in [ALWZ ’07].
This is the only scaling leading to a continuum limit qualitatively
different from DT and CDT.

61. The number of local maxima Nmax (a) scales with N at the critical
point,
Nmax (a) N
N
= 2
a
2
2/3
+ O(a),
Nmax (a = 1) N
N
= 1/2
Therefore, to obtain a finite continuum density of critical points one
should scale a ∝ N−3/2
, i.e. a = gs
3
as observed in [ALWZ ’07].
This is the only scaling leading to a continuum limit qualitatively
different from DT and CDT.
Continuum limit g = gc (a)(1 − Λ 2
),
z− = z−,c (1 − Z ), a = gs
3
:
Z3
− Λ + 3
gs
2
2/3
Z − gs = 0

62. The number of local maxima Nmax (a) scales with N at the critical
point,
Nmax (a) N
N
= 2
a
2
2/3
+ O(a),
Nmax (a = 1) N
N
= 1/2
Therefore, to obtain a finite continuum density of critical points one
should scale a ∝ N−3/2
, i.e. a = gs
3
as observed in [ALWZ ’07].
This is the only scaling leading to a continuum limit qualitatively
different from DT and CDT.
Continuum limit g = gc (a)(1 − Λ 2
),
z− = z−,c (1 − Z ), a = gs
3
:
Z3
− Λ + 3
gs
2
2/3
Z − gs = 0
“Cup function” WΛ(X) = 1
X+Z . Agrees with
[ALWZ ’07].

63. Two loop identity in generalized CDT
Consider surfaces with two boundaries separated
by a geodesic distance D.

64. Two loop identity in generalized CDT
Consider surfaces with two boundaries separated
by a geodesic distance D.
One can assign time T1, T2 to the boundaries
(|T1 − T2| ≤ D) and study a “merging” process.

65. Two loop identity in generalized CDT
Consider surfaces with two boundaries separated
by a geodesic distance D.
One can assign time T1, T2 to the boundaries
(|T1 − T2| ≤ D) and study a “merging” process.
For a given surface the foliation depends on
T1 − T2, hence also Nmax and its weight.

66. Two loop identity in generalized CDT
Consider surfaces with two boundaries separated
by a geodesic distance D.
One can assign time T1, T2 to the boundaries
(|T1 − T2| ≤ D) and study a “merging” process.
For a given surface the foliation depends on
T1 − T2, hence also Nmax and its weight.
However, in [ALWZ ’07] it was shown that the
amplitude is independent of T1 − T2.

67. Two loop identity in generalized CDT
Consider surfaces with two boundaries separated
by a geodesic distance D.
One can assign time T1, T2 to the boundaries
(|T1 − T2| ≤ D) and study a “merging” process.
For a given surface the foliation depends on
T1 − T2, hence also Nmax and its weight.
However, in [ALWZ ’07] it was shown that the
amplitude is independent of T1 − T2.
Refoliation symmetry at the quantum level in the
presence of topology change!

68. Two loop identity in generalized CDT
Consider surfaces with two boundaries separated
by a geodesic distance D.
One can assign time T1, T2 to the boundaries
(|T1 − T2| ≤ D) and study a “merging” process.
For a given surface the foliation depends on
T1 − T2, hence also Nmax and its weight.
However, in [ALWZ ’07] it was shown that the
amplitude is independent of T1 − T2.
Refoliation symmetry at the quantum level in the
presence of topology change!
Can we better understand this symmetry at the
discrete level?

69. Two loop identity in generalized CDT
Consider surfaces with two boundaries separated
by a geodesic distance D.
One can assign time T1, T2 to the boundaries
(|T1 − T2| ≤ D) and study a “merging” process.
For a given surface the foliation depends on
T1 − T2, hence also Nmax and its weight.
However, in [ALWZ ’07] it was shown that the
amplitude is independent of T1 − T2.
Refoliation symmetry at the quantum level in the
presence of topology change!
Can we better understand this symmetry at the
discrete level?
For simplicity set the boundarylengths to zero.
Straightforward generalization to finite boundaries.

70. T1 = 0
T2 = 0
D = 4

71. T1 = 0
T2 = 0
D = 4

72. t t
t 1
t 1
t t
t 1
t 1
T1 = 0
T2 = 0
D = 4

73. t t
t 1
t 1
t t
t 1
t 1
T1 = 0
T2 = 0
D = 4

74. t t
t 1
t 1
t t
t 1
t 1
T1 = 0
T2 = 0
D = 4

75. 8
7
7
7
7
9
9
7
6
8
8
68
6
6
6
8
6
8
8
5
5
7
5
5
7
5
7
7
5
7
5
5
7
5
7
5
7
6
6
6
4
6
4
4
4
4
6
4
4
4
6
4 4
6
4
6
6
6
3
5
3
5
5
5
5
5
3
3
5
3
5
5
5
3
3
5
3
5
3
3
4
2
6
2
2
4
4
4
4
2
2
4
2
4
2
4
4
4
4 4
4
2
1
1
3
3
5
5
3
3
5
3
1
3
3
3
3
5
5
3
3
3
4
0
2
4
2
4
2
2
2
4
4
4
2
2
4
2
4
2
3
3
3
1
3
3
1
1
3
1
3
3
3
3
0
2
2
2
2
2
2
2
1
t t
t 1
t 1
t t
t 1
t 1
T1 = 0
T2 = 0
D = 4

76. 8
7
7
7
7
7
6
8
68
6
6
6
6
8
5
5
7
5
5
7
5
7
7
5
7
5
5
7
5
7
5
7
6
6
6
4
6
4
4
4
4
6
4
4
4
6
4 4
6
4
6
6
6
3
5
3
5
5
5
5
5
3
3
5
3
5
5
5
3
3
5
3
5
3
3
4
2
2
2
4
4
2
2
4
2
4
2
4
4
4
4 4
4
2
1
1
3
3
5
5
3
3
3
1
3
3
3
3
5
5
3
3
3
4
0
2
2
4
2
2
2
4
4
4
2
2
2
2
3
3
3
1
3
1
1
3
1
3
3
0
2
2
2
2
2
2
1
t t
t 1
t 1
t t
t 1
t 1
T1 = 0
T2 = 0
D = 4
No local minima ⇒ the labeling is canonical!

77. 0
0
t t
t 1
t 1
t t
t 1
t 1
T1 = 0
T2 = 0
D = 4
No local minima ⇒ the labeling is canonical!

78. 0
2
t t
t 1
t 1
t t
t 1
t 1
T1 = 0
T2 = 2
D = 4
No local minima ⇒ the labeling is canonical!

79. 8
7
7
7
7
7
6
9
69
6
6
6
6
8
5
5
7
5
5
8
5
8
9
5
7
5
5
7
5
8
5
9
7
8
8
4
6
4
4
4
4
8
4
4
4
8
4 4
7
4
8
8
8
3
7
3
7
7
7
7
7
3
3
7
3
5
6
7
3
3
7
3
7
3
3
6
2
2
2
6
6
2
2
4
2
6
2
6
6
6
6 6
6
2
1
1
5
5
7
5
5
3
4
1
5
5
5
4
6
7
5
5
3
4
0
2
4
6
4
2
3
5
6
6
4
2
4
2
5
5
3
3
3
3
3
5
1
5
4
2
4
4
2
4
4
4
3
t t
t 1
t 1
t t
t 1
t 1
T1 = 0
T2 = 2
D = 4
No local minima ⇒ the labeling is canonical!
There exists a bijection preserving the number of local maxima:
T1,T2
←→
0
0
T1,T2
←→
0
2
T1 ,T2
←→
T1 ,T2

80. Conclusions & Outlook
Conlusions
The Cori–Vauquelin–Schaeffer bijection encodes 2d geometry in
trees, which are simple objects from an analytical point of view.
The bijection is ideal for studying “proper-time foliations” of random
surfaces.
In the setting of generalized CDT, the bijection exposes symmetries
in certain amplitudes with prescibed time on the boundaries.

81. Conclusions & Outlook
Conlusions
The Cori–Vauquelin–Schaeffer bijection encodes 2d geometry in
trees, which are simple objects from an analytical point of view.
The bijection is ideal for studying “proper-time foliations” of random
surfaces.
In the setting of generalized CDT, the bijection exposes symmetries
in certain amplitudes with prescibed time on the boundaries.
What I’ve not shown (see our forthcoming paper)
Similar bijections exist for triangulations, but more involved.
Explicit expressions can be derived for transition amplitudes
G(L1, L2; T) in generalized CDT.

82. Conclusions & Outlook
Conlusions
The Cori–Vauquelin–Schaeffer bijection encodes 2d geometry in
trees, which are simple objects from an analytical point of view.
The bijection is ideal for studying “proper-time foliations” of random
surfaces.
In the setting of generalized CDT, the bijection exposes symmetries
in certain amplitudes with prescibed time on the boundaries.
What I’ve not shown (see our forthcoming paper)
Similar bijections exist for triangulations, but more involved.
Explicit expressions can be derived for transition amplitudes
G(L1, L2; T) in generalized CDT.
Outlook
What is the exact structure of these symmetries? Related to
conformal symmetry (Virasoro algebra)?
Straightforward extension to higher genus. Sum over topologies in
non-critical string theory?

83. Merry Christmas!

84. Appendix: canonical labeling
0
2

85. Appendix: canonical labeling
1
1
1
0
1
2

86. Appendix: canonical labeling
2
2
2
2
2
2
2
2
1
1
1
0
2
2
2
2
1
2
2

87. Appendix: canonical labeling
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
2
2
2
1
1
3
1
3
0
2
2
3
2
2
3
3
3
3
3
1
2
2
3

88. Appendix: canonical labeling
4
4
4
4
4
4
4
4
4 4
4
3
3
3
3
3
3
3
3
3
3
2
2
2
2
2
4
2
2
2
1
1
3
4
1
43
4
0
2
4
4
2
3
4
2
4
2
3
3
3
3
3
1 4
2
4
4
2
4
4
4
3

89. Appendix: canonical labeling
5
5
5
5
5
5
5
5
5
5
4
4
4
4
4
4
4
4
4 4
4
3
3
3
3
3
5
3
3
3
3
3
2
2
2
2
2
4
2
2
2
1
1
5
5
5
5
3
4
1
5
5
5
4
5
5
3
4
0
2
4
4
2
3
5
4
2
4
2
5
5
3
3
3
3
3
5
1
5
4
2
4
4
2
4
4
4
3

90. Appendix: canonical labeling
6
6
6
6
6
6
5
5
5
5
5
5
5
5
5
5
4
6
4
4
4
4
4
4
4
4 4
4
3
3
3
3
3
5
6
3
3
3
3
3
6
2
2
2
6
6
2
2
4
2
6
2
6
6
6
6 6
6
2
1
1
5
5
5
5
3
4
1
5
5
5
4
6
5
5
3
4
0
2
4
6
4
2
3
5
6
6
4
2
4
2
5
5
3
3
3
3
3
5
1
5
4
2
4
4
2
4
4
4
3

91. Appendix: canonical labeling
7
7
7
7
7
6
6
6
6
6
6
5
5
7
5
5
5
5
7
5
5
7
5
5
7
4
6
4
4
4
4
4
4
4
4 4
7
4
3
7
3
7
7
7
7
7
3
3
7
3
5
6
7
3
3
7
3
7
3
3
6
2
2
2
6
6
2
2
4
2
6
2
6
6
6
6 6
6
2
1
1
5
5
7
5
5
3
4
1
5
5
5
4
6
7
5
5
3
4
0
2
4
6
4
2
3
5
6
6
4
2
4
2
5
5
3
3
3
3
3
5
1
5
4
2
4
4
2
4
4
4
3

92. Appendix: canonical labeling
8
7
7
7
7
7
6
6
6
6
6
6
8
5
5
7
5
5
8
5
8
5
7
5
5
7
5
8
5
7
8
8
4
6
4
4
4
4
8
4
4
4
8
4 4
7
4
8
8
8
3
7
3
7
7
7
7
7
3
3
7
3
5
6
7
3
3
7
3
7
3
3
6
2
2
2
6
6
2
2
4
2
6
2
6
6
6
6 6
6
2
1
1
5
5
7
5
5
3
4
1
5
5
5
4
6
7
5
5
3
4
0
2
4
6
4
2
3
5
6
6
4
2
4
2
5
5
3
3
3
3
3
5
1
5
4
2
4
4
2
4
4
4
3

93. Appendix: canonical labeling
8
7
7
7
7
7
6
9
69
6
6
6
6
8
5
5
7
5
5
8
5
8
9
5
7
5
5
7
5
8
5
9
7
8
8
4
6
4
4
4
4
8
4
4
4
8
4 4
7
4
8
8
8
3
7
3
7
7
7
7
7
3
3
7
3
5
6
7
3
3
7
3
7
3
3
6
2
2
2
6
6
2
2
4
2
6
2
6
6
6
6 6
6
2
1
1
5
5
7
5
5
3
4
1
5
5
5
4
6
7
5
5
3
4
0
2
4
6
4
2
3
5
6
6
4
2
4
2
5
5
3
3
3
3
3
5
1
5
4
2
4
4
2
4
4
4
3