Trees
Definition
An undirected graph is a tree if it is connected and acyclic. An undirected graph is a forest if it is acyclic. Each connected component of a forest is a tree, motivating the definition.
Properties
There is a unique path between any two
vertices of a tree.
Such a path exists because the tree is
connected.
Such a path is unique because the existence
of two separate paths would create a cycle.
Distance
The distance between two vertices $v$ and $w$ in a tree is the length of the unique path connnecting $v$ and $w$. Recall that the length of a path is the number of edges, or one fewer than the number of vertices. If $v$ and $w$ are adjacent in the tree, then there distance is 1.