  Edge connectivity of a graph is one of the most fundamental graph-theoretic
concepts. The celebrated tree packing theorem of Tutte and Nash-Williams from
1961 states that every $k$-edge connected graph $G$ contains a collection
$\cal{T}$ of $\lfloor k/2 \rfloor$ edge-disjoint spanning trees, that we refer
to as a tree packing; the diameter of the tree packing $\cal{T}$ is the largest
diameter of any tree in $\cal{T}$. A desirable property of a tree packing, that
is both sufficient and necessary for leveraging the high connectivity of a
graph in distributed communication, is that its diameter is low. Yet, despite
extensive research in this area, it is still unclear how to compute a tree
packing, whose diameter is sublinear in $|V(G)|$, in a low-diameter graph $G$,
or alternatively how to show that such a packing does not exist.
  In this paper we provide first non-trivial upper and lower bounds on the
diameter of tree packing. First, we show that, for every $k$-edge connected
$n$-vertex graph $G$ of diameter $D$, there is a tree packing $\cal{T}$ of size
$\Omega(k)$, diameter $O((101k\log n)^D)$, that causes edge-congestion at most
$2$. Second, we show that for every $k$-edge connected $n$-vertex graph $G$ of
diameter $D$, the diameter of $G[p]$ is $O(k^{D(D+1)/2})$ with high
probability, where $G[p]$ is obtained by sampling each edge of $G$
independently with probability $p=\Theta(\log n/k)$. This provides a packing of
$\Omega(k/\log n)$ edge-disjoint trees of diameter at most $O(k^{(D(D+1)/2)})$
each. We then prove that these two results are nearly tight. Lastly, we show
that if every pair of vertices in a graph has $k$ edge-disjoint paths of length
at most $D$ connecting them, then there is a tree packing of size $k$, diameter
$O(D\log n)$, causing edge-congestion $O(\log n)$. We also provide several
applications of low-diameter tree packing in distributed computation.
