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In this paper we demonstrate connections between three seemingly unrelatedconcepts.  (1) The discrete isoperimetric problem in the infinite binary tree with allthe leaves at the same level, $ {\mathcal T}_{\infty}$:  The $n$-th edge isoperimetric number $\delta(n)$ is defined to be$\min_{|S|=n, S \subset V({\mathcal T}_{\infty})} |(S,\bar{S})|$, where$(S,\bar{S})$ is the set of edges in the cut defined by $S$.  (2) Signed almost binary partitions: This is the special case of thecoin-changing problem where the coins are drawn from the set  ${\pm (2^d - 1): $d$ is a positive integer}$. The quantity of interest is$\tau(n)$, the minimum number of coins necessary to make change for $n$ cents.  (3) Certain Meta-Fibonacci sequences: The Tanny sequence is defined by$T(n)=T(n{-}1{-}T(n{-}1))+T(n{-}2{-}T(n{-}2))$ and the Conolly sequence isdefined by $C(n)=C(n{-}C(n{-}1))+C(n{-}1{-}C(n{-}2))$, where the initialconditions are $T(1) = C(1) = T(2) = C(2) = 1$.  These are well-known "meta-Fibonacci" sequences.  The main result that ties these three together is the following: $$ \delta(n)= \tau(n) = n+ 2 + 2 \min_{1 \le k \le n} (C(k) - T(n-k) - k).$$ Apart fromthis, we prove several other results which bring out the interconnectionsbetween the above three concepts.

Isoperimetric Sequences for Infinite Complete Binary Trees, Meta-Fibonacci Sequences and Signed Almost Binary Partitions