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Balanced rooted trees
42 · 420
Unlabeled unordered rooted trees with every leaf at the same height. OEIS A048816. a(10)=42. a(15)=420.
A tree starts at one point and branches. Here every leaf must arrive at the same depth. With ten nodes there are exactly forty-two such shapes, if you forget labels and forget order around a fork. With fifteen nodes there are four hundred twenty. One sequence. Two exact counts. Nobody wrote the rule to find them.
What it supports
OEIS A048816. Christian G. Bower, 15 April 1999. Number of unlabeled unordered rooted trees with n nodes and every leaf at the same height. Offset 1. Data: 1, 1, 2, 3, 5, 7, 12, 17, 28, 42, 68, 103, 168, 260, 420. So a(10)=42 and a(15)=420. Joerg Arndt lists the ten-node shapes. A 2025 Journal of Integer Sequences paper on quasi-full t-ary trees cites the sequence. That paper counts a finer family.
What it does not
Not pronic 6×7. Not 42×10. Not Catalan C₅. Not p(10)=42. Those are other questions. The pairing of 42 with 420 is seen after the count. Quasi-full t-ary trees (A352460) sum to 41 at ten nodes, not 42. They are a different count.
Calculation
A048816 offset 1. a(10)=42. a(15)=420. n = number of nodes.
Unlabeled unordered rooted trees. Every leaf at the same height. Enumeration, not a search for 42.
ExactnessExact
Form420
Unittrees
Strengthhard
When1999
Hung15 September 2026