An important topic of discrete Mathematics. #M_ARY_TREE.

1. M–ary Tree
Presented By:
Md. Arif Hasan – BSSE1112
Tahmeed Mahbub – BSSE1127
Tasmia Zerin – BSSE1128

2. Balanced
Tree
B
Tree
General
Tree
M – ary
Tree
Where is M-ary
tree in the tree of
TREEs ?
AVL
Tree
Red/
Black
Tree
B+
Tree
Here I am!
Binary
Tree
1.

3. What is M-ary Tree?
A rooted tree is called an m-ary tree if every internal vertex has no more then m
children.
Types of m-ary tree:
• A full m-ary tree is an m-ary tree where within each level
every node has either 0 or m children.
• A complete m-ary tree is an m-ary tree which must be
completely filled on every level except for the last level.
However, if the last level is not complete, then all nodes
of the tree must be as far left as possible.
• A perfect m-ary tree is a full m-ary tree in which all leaf
nodes are at the same depth.
1 2 3 m
Internal Vertices
2.

4. Properties of m-ary tree:
• A full m-ary tree with i internal vertices contains n= mi + 1
vertices and l=(m-1)i+1 leaves
i = 4 internal vertices
m = 3
n = 3 x 4 + 1 = 13 vertices, as 12
vertices (except the root) are children of
those 4 internal vertices
So, children = m x i = 4 x 3 = n – 1
n = l + i => l = (m-1)i + 1
l = 2 x 4 + 1 = 9 leaves
Internal Vertices
3.

5. The children of m-ary tree:
Binary
Tree
M – ary
Tree
Ternar
y Tree
M = 2 M = 3
Yes, I’m the parent of all 1,2,3……
ary trees
4.

6. Traversal Methods for m-ary tree:
Traversing a m-ary tree is very similar to binary tree traversal. But since there are
more than two children per node for m > 2, one must define the notion
of left and right subtrees. One common method to establish left/right subtrees is
to divide the list of children nodes into two groups.By defining an order on the
m children of a node, the first {0,1,…. └m/2┘} nodes would constitute the left
subtree and {
┌m/2
┐,…..,m} would constitute the right subtree.
5.

7. Left Subtree Right Subtree
Here, for m = 3,
left subtree will
be { 0, 1 } (as 3/2 =
1) and right
subtree will be { 2
}
For Example:
6.

8. Methods of Storing m-ary trees:
• Arrays: if the tree is a
complete m-ary tree, this
method wastes no space. In
this compact arrangement, if
a node has an index i, its c-
th child in range {1,...,m} is
found at index m.i + c. The
space complexity of this
method is O ( mn )
A
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0
1 2 3
4 5 6 7 8 9 10 11 12
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9. • Pointer Based: Each node
would have an internal array
for storing pointers to each of
its m children. Compared to
array-based implementation,
this implementation method
has superior space complexity
of O (m.n)
8.

10. Convert a m-ary tree to binary tree:
Converting an arbitrary m-ary tree to a binary tree would only increase the height of
the tree by a constant factor and would not affect the overall worst-case time
complexity. First, we link all the immediate children nodes of a given parent node
together in order to form a link list. Then, we keep the link from the parent to the
first (i.e., the leftmost) child and remove all the other links to the rest of the children.
We repeat this process for all the children (if they have any children) until we have
processed all the internal nodes and rotate the tree by 45 degrees clockwise. The tree
obtained is the desired binary tree obtained from the given m-ary tree.
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Converting m-ary to binary tree
10.

12. Thank You