1. Trees Recall that a tree is a connected acyclic graph. Note that this is similar to but slightly
different from a rooted tree that you might have come across before in the Context of data
structures. In our context the tree does not have any designated root vertex, and is just a graph on
n vertices and with m edges. We start by proving two useful properties of trees: (C) If for ever
pair of vertices in a graph they are connected by exactly one simple path. then the graph must be
a tree. (d) If the graph has no simple cycles but has the property that the addition of any single
edge (not already in the graph) will create a simple cycle, then the graph is a tree.
Solution
1-
Since T is a tree, then there is a unique path connecting any two distinct vertices of T
Hence, there are () distinct paths in T
2-
Suppose T is a tree, then by de?nition of a tree it has no simple circuits. Now in T let us
Add an edge e which connects any two non-adjacent vertices u and v. Adding this edge
De?nitely produce a circuit because the resulting graph now has more edges to be a tree.
“Oh GOSH! Reflecting on Hackteria's Collaborative Practices in a Global Do-It...
1. Trees Recall that a tree is a connected acyclic graph. Note that t.pdf
1. 1. Trees Recall that a tree is a connected acyclic graph. Note that this is similar to but slightly
different from a rooted tree that you might have come across before in the Context of data
structures. In our context the tree does not have any designated root vertex, and is just a graph on
n vertices and with m edges. We start by proving two useful properties of trees: (C) If for ever
pair of vertices in a graph they are connected by exactly one simple path. then the graph must be
a tree. (d) If the graph has no simple cycles but has the property that the addition of any single
edge (not already in the graph) will create a simple cycle, then the graph is a tree.
Solution
1-
Since T is a tree, then there is a unique path connecting any two distinct vertices of T
Hence, there are () distinct paths in T
2-
Suppose T is a tree, then by de?nition of a tree it has no simple circuits. Now in T let us
Add an edge e which connects any two non-adjacent vertices u and v. Adding this edge
De?nitely produce a circuit because the resulting graph now has more edges to be a tree
