Induct on the number of edges. If it has one edge it must have 2 nodes, one at each end of the edge.
Assume true for all trees with edges. Then the number of nodes is on such a tree.
Let be a tree with edges. Then it contains a subtree with edges (just delete one of the bottom-most edges and the ONE node at its tip). This tree, , has nodes, but as we deleted ONE node from this means that must have nodes.
As we have that as required.
Is there anything you specifically do not understand in this?...