Let there be a tree with at least two vertices. One vertex has degree $k$ and the other has degree $l$. Prove that such tree has at least $k + l - 2$ leaves.

My logic is that from one vertex you can reach at least $k$ leaves and from the other vertex you can reach at least $l$ leaves. But how exactly would I prove that you have to subtract $2$? Is it because if you start from one of the given vertices and go through the second one, then one of the leaves will be a duplicate?

  • $\begingroup$ If you start from one of the vertices and follow a path of vertices to the second, you have discovered a cycle. And cycles are not allowed in trees. $\endgroup$ – Matthew Leingang May 13 at 18:00

In the tree $T$ there's a path from the vertex $v$ of degree $k$ to the vertex $w$ of degree $l$. Let this path start with the edge $e$. There are $k-1$ other edges from $v$, and one can walk from $v$ starting at any of these edges and reach a leaf. This accounts for $k-1$ leaves. Likewise start at $w$ to reach $l-1$ further leaves.

  • $\begingroup$ A relevant (I think) addition is that the path from $v$ to $w$ is unique in a tree. So OP's concern about following a branch along an edge other than $e$ from $v$ and accidentally ending up at $w$ won't happen. $\endgroup$ – Matthew Leingang May 13 at 18:04

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