Prove by induction that there is a unique path between any p

Prove by induction that there is a unique path between any pair of vertices in a tree.

Solution

Each node here the tree has a parent. The tree should have single node. Hence the entire nodes must be attached.

Base Case: A graph among 1 root R and 2 child nodes X and Y. An argument can be ready for now 1 child, still. In such case unique trail will exist : X R Y

Inductive Case: Let present 2 nodes A and B, such to unique pathway exist between them. A has children a1, a2. B has children b1, b2. A pathway must exist among parent and child. Therefore path should exist among the pairs: (a1, a2), (b1, b2). (a1, B), (A,b1)