  For fixed $t\ge 2$, we consider the class of representations of $1$ as sum of
unit fractions whose denominators are powers of $t$ or equivalently the class
of canonical compact $t$-ary Huffman codes or equivalently rooted $t$-ary plane
"canonical" trees. We study the probabilistic behaviour of the height (limit
distribution is shown to be normal), the number of distinct summands (normal
distribution), the path length (normal distribution), the width (main term of
the expectation and concentration property) and the number of leaves at maximum
distance from the root (discrete distribution).
