  Local convergence of bounded degree graphs was introduced by Benjamini and
Schramm. This result was extended further by Lyons to bounded average degree
graphs. In this paper we study the convergence of random tree sequences with
given degree distributions. Denote by ${\cal D}_n$ the set of possible degree
sequences of a tree on $n$ nodes. Let ${\bm D}_n$ be a random variable on
${\cal D}_n$ and ${\bm T}({\bm D}_n)$ be a uniform random tree with degree
sequence ${\bm D}_n$. We show that the sequence ${\bm T}({\bm D}_n)$ converges
in probability if and only if ${\bm D}_n\to {\bm D}=({\bm D}(i))_{i=1}^\infty$,
where ${\bm D}(i)\sim {\bm D}(j)$, $\mathds{E}({\bm D}(1))=2$ and ${\bm D}(1)$
is a random variable on $\mathds{N}^+$.
