Pith Number

pith:TS2HDKBY

pith:2026:TS2HDKBYXSCBEFHZIS56VGEMEC

not attested not anchored not stored refs resolved

The Distribution of the Deepest Leaves in Binary Trees

Antoine Genitrini, Khaydar Nurligareev, Olivier Bodini

In random plane binary trees, the number of leaves at maximum depth has a limiting distribution with probability of exactly 2m such leaves asymptotically equal to 4^{-m+1}.

arxiv:2605.12821 v1 · 2026-05-12 · math.CO

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Claims

C1strongest claim

The probability of having 2m deepest leaves satisfies κ[m] ∼ 4^{-m+1}; the limiting distribution is characterized by a functional equation obtained via the Fatou coordinate of an Abel equation.

C2weakest assumption

The truncation error in the critical scaling regime of the Catalan iteration admits a three-zone dominated-convergence analysis that produces clean square-root singular expansions for the associated generating functions.

C3one line summary

In random binary trees the number of deepest leaves follows a distribution with mean approximately 2.8037, P(exactly two) approximately 0.7009, and tail probability decaying as 4 to the minus m.

References

18 extracted · 18 resolved · 0 Pith anchors

[1] Bodini, O. and Genitrini, A. and Nurligareev, K. , title =. 2026 , howpublished = 2026

[2] Aldous, D. , title =. Ann. Probab. , volume =

[3] Aldous, D. , title =. Stochastic Analysis: Proceedings of the Durham Symposium on Stochastic Analysis, 1990 , publisher=. 1991 , pages= 1990

[4] Colloquium on trees in algebra and programming , pages= 1992

[5] Bodini, O. and Genitrini, A. and Gittenberger, B. and Wagner, S. , title =. Discrete Math. , volume =

Receipt and verification

First computed	2026-05-18T03:09:12.217677Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

9cb471a838bc841214f944bbea988c20bb32cdc8e1d932846e77053ccebb6cc0

Aliases

arxiv: 2605.12821 · arxiv_version: 2605.12821v1 · doi: 10.48550/arxiv.2605.12821 · pith_short_12: TS2HDKBYXSCB · pith_short_16: TS2HDKBYXSCBEFHZ · pith_short_8: TS2HDKBY

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/TS2HDKBYXSCBEFHZIS56VGEMEC \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 9cb471a838bc841214f944bbea988c20bb32cdc8e1d932846e77053ccebb6cc0

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "2472adfc74dc13906941ad07b283a0e43c6403cbeab712cd8715026e0f5a047c",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-12T23:36:28Z",
    "title_canon_sha256": "9806d951658f696c46624169dbf95ada73aabbf5f5c041b6df840957edd804d9"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.12821",
    "kind": "arxiv",
    "version": 1
  }
}