  We prove an analogue for trees of Courant's theorem on the interlacing
property of zeros of eigenfunctions of a Schr\"{o}dinger operator. Let $\Gamma$
be a finite tree, and $\mathcal A$ a Schr\"{o}dinger operator on $\Gamma$. If
the eigenvectors of $\mathcal A$ are ordered according to increasing
eigenvalues, and the vertices corresponding to zero coordinates are of degree
at most two, then the zeros of the linear extensions of eigenvectors have the
interlacing property.
