Intuitive proof for a tree with n nodes, has n-1 edges
You can see a (binary) tree as a directed graph: suppose the root is the "lowest" node and the leaves are the "highest" ones, then say that all the edges are oriented upwards. Then, every node that is not the root will have exactly one edge entering in it, and every edge will be pointing at exactly one node. This means that if you have $n$ nodes, you have to have $n-1$ edges (one per node with the exception of the root).