Let $T$ be a tree with 10 vertices. The sum of the degrees of all the vertices in $T$ is ________.

1. A tree with $n$ vertices always contains exactly $e = n - 1$ edges. For $n = 10$:

   $$e = 10 - 1 = 9$$

2. By the Handshaking Lemma (degree sum formula), the sum of the degrees of all vertices in any undirected graph is twice the number of edges:

   $$\sum_{v \in V} \deg(v) = 2e = 2 \times 9 = 18$$

Thus, the sum of the degrees of all the vertices in $T$ is 18.