  Consider the computations at a node in a message passing algorithm. Assume
that the node has incoming and outgoing messages $\mathbf{x} = (x_1, x_2,
\ldots, x_n)$ and $\mathbf{y} = (y_1, y_2, \ldots, y_n)$, respectively. In this
paper, we investigate a class of structures that can be adopted by the node for
computing $\mathbf{y}$ from $\mathbf{x}$, where each $y_j, j = 1, 2, \ldots, n$
is computed via a binary tree with leaves $\mathbf{x}$ excluding $x_j$. We make
three main contributions regarding this class of structures. First, we prove
that the minimum complexity of such a structure is $3n - 6$, and if a structure
has such complexity, its minimum latency is $\delta + \lceil \log(n-2^{\delta})
\rceil$ with $\delta = \lfloor \log(n/2) \rfloor$, where the logarithm always
takes base two. Second, we prove that the minimum latency of such a structure
is $\lceil \log(n-1) \rceil$, and if a structure has such latency, its minimum
complexity is $n \log(n-1)$ when $n-1$ is a power of two. Third, given $(n,
\tau)$ with $\tau \geq \lceil \log(n-1) \rceil$, we propose a construction for
a structure which we conjecture to have the minimum complexity among structures
with latencies at most $\tau$. Our construction method runs in $O(n^3
\log^2(n))$ time, and the obtained structure has complexity at most (generally
much smaller than) $n \lceil \log(n) \rceil - 2$.
