Question

Can there be two different simple paths between the vertices of a tree?

Short answer

No, there could not be 2 different simple paths between the vertices of tree.

Step-by-step solution

Definition

A simple path is a path that does not contain the same edge more than once.

A circuit is a path that begins and ends in the same vertex.

A graph is connected if there exists a path between every pair of vertices.

A tree is an undirected graph that is connected and that does not contain any simple circuits.

Obtaining the graph

Let \({\bf{u}}\) and \({\bf{v}}\) be two different vertices of the graph \({\bf{T}}\).

If \({\bf{T}}\) is a tree, then \({\bf{T}}\) can only contain a unique simple path from \({\bf{u}}\) to \({\bf{v}}\), because if there are two unique simple paths from \({\bf{u}}\) to \({\bf{v}}\) that combining these two simple paths will result in a path that contains some simple circuit.