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I'm new in discrete math. Can someone prove simply that a tree with $n$ nodes must have exactly $n - 1$ edges. I have researched the solution but I haven't founded yet. I know of course, a tree with n nodes must have exactly n - 1 edges. But, I can't prove it. Thank you. induction on $n$

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It's easy to see that any tree with $1$ node has $0$ edges. Now let's assume that any tree with $k$ nodes has $k-1$ edges. Now, given any tree with $k+1$ nodes there must exist at least one leaf. Remove a leaf and the connecting edge. The resulting tree has $k$ nodes and by our induction assumption it has $k-1$ edges. Hence, the tree with $k+1$ nodes must have $k$ edges. And from induction, we have the result.

chandu1729
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  • I think a slightly difficult technical point is to prove that every tree with more than $1$ node has at least one leaf (in fact it has at least $2$ leaves). It is intuitively obvious, but that does not really help. – Marc van Leeuwen Jan 25 '15 at 19:41
  • If there is no leaf, then there must be cycle and it is not permitted for trees! – chandu1729 Jan 25 '15 at 19:55
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    @chandu1729: That is indeed a way to prove that every tree must have a leaf, but the argument still has to be made. – Brian M. Scott Jan 25 '15 at 20:14