Number of maximum degree vertices in a tree Let $T$ be a tree of order $n$, where $T$ has $\Delta(T) \geq 3$ leaves. Show that $T$ has exactly one vertex of maximum degree $\Delta(T)$.
I am stuck on how to prove this statement. 

Suppose that there is 2 or more vertices of maximum degree $\Delta (T)$, then the number of leaves will be at least $\Delta (T) -2$, since (I'll consider it a directed graph now just to use the terms indegree and out degree, but it also works in ordinary graphs) every vertex in a tree has its indegree $1$ or $0$, so its out degree is at least $\Delta (T)-1$, and each edge coming OUT of this vertex is connected to at least $1$ leaf. (2 vertices can't be connected to the same leaf also).
