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<p>A normal (phylogenetic) network with <i>k</i> reticulations displays <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(2^k\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mi>k</mi> </msup> </math></EquationSource> </InlineEquation> phylogenetic trees. In this paper, we establish an analogous result for tree-child (phylogenetic) networks with no underlying 3-cycles. In particular, we show that a tree-child network with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(k\ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> reticulations and no underlying 3-cycles displays at least <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(2^{{k}/{2}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> phylogenetic trees if <i>k</i> is even and at least <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\frac{3}{2\sqrt{2}}2^{{k}/{2}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>3</mn> <mrow> <mn>2</mn> <msqrt> <mn>2</mn> </msqrt> </mrow> </mfrac> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> if <i>k</i> is odd. Moreover, we show that these bounds are sharp and characterise the tree-child networks that attain these bounds.</p>

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A Sharp Lower Bound for the Number of Phylogenetic Trees Displayed by a Tree-Child Network

  • 期刊论文

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  • 发表日期：

    2026年5月13日

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  • Charles Semple,
  • Kristina Wicke

摘要

A normal (phylogenetic) network with k reticulations displays \(2^k\) 2 k phylogenetic trees. In this paper, we establish an analogous result for tree-child (phylogenetic) networks with no underlying 3-cycles. In particular, we show that a tree-child network with \(k\ge 2\) k ≥ 2 reticulations and no underlying 3-cycles displays at least \(2^{{k}/{2}}\) 2 k / 2 phylogenetic trees if k is even and at least \(\frac{3}{2\sqrt{2}}2^{{k}/{2}}\) 3 2 2 2 k / 2 if k is odd. Moreover, we show that these bounds are sharp and characterise the tree-child networks that attain these bounds.