Number of trees

These functions return the number of rooted or unrooted binary trees consistent with a given pattern of splits.

NRooted(tips)

NUnrooted(tips)

NRooted64(tips)

NUnrooted64(tips)

LnUnrooted(tips)

LnUnrooted.int(tips)

Log2Unrooted(tips)

Log2Unrooted.int(tips)

LnRooted(tips)

LnRooted.int(tips)

Log2Rooted(tips)

Log2Rooted.int(tips)

LnUnrootedSplits(...)

Log2UnrootedSplits(...)

NUnrootedSplits(...)

LnUnrootedMult(...)

Log2UnrootedMult(...)

NUnrootedMult(...)

Arguments

tips

Integer specifying the number of leaves.

...

Integer vector, or series of integers, listing the number of leaves in each split.

Details

Functions starting N return the number of rooted or unrooted trees. Replace this initial N with Ln for the natural logarithm of this number; or Log2 for its base 2 logarithm.

Calculations follow Cavalli-Sforza and Edwards (1967) and Carter et al. (1990) , Theorem 2.

Functions

• NUnrooted(): Number of unrooted trees

• NRooted64(): Exact number of rooted trees as 64-bit integer (13 < nTip < 19)

• NUnrooted64(): Exact number of unrooted trees as 64-bit integer (14 < nTip < 20)

• LnUnrooted(): Log Number of unrooted trees

• LnUnrooted.int(): Log Number of unrooted trees (as integer)

• LnRooted(): Log Number of rooted trees

• LnRooted.int(): Log Number of rooted trees (as integer)

• NUnrootedSplits(): Number of unrooted trees consistent with a bipartition split.

• NUnrootedMult(): Number of unrooted trees consistent with a multi-partition split.

References

Carter M, Hendy M, Penny D, Székely LA, Wormald NC (1990). “On the distribution of lengths of evolutionary trees.” SIAM Journal on Discrete Mathematics, 3(1), 38--47. doi:10.1137/0403005 .

Cavalli-Sforza LL, Edwards AWF (1967). “Phylogenetic analysis: models and estimation procedures.” Evolution, 21(3), 550--570. ISSN 00143820, doi:10.1111/j.1558-5646.1967.tb03411.x .

See also

Other tree information functions: CladisticInfo(), TreesMatchingTree()

Author

Martin R. Smith (martin.smith@durham.ac.uk)

Examples

NRooted(10)
#> [1] 34459425
NUnrooted(10)
#> [1] 2027025
LnRooted(10)
#> [1] 17.35529
LnUnrooted(10)
#> [1] 14.52208
Log2Unrooted(10)
#> [1] 20.95093
# Number of trees consistent with a character whose states are
# 00000 11111 222
NUnrootedMult(c(5,5,3))
#> [1] 694575

NUnrooted64(18)
#> integer64
#> [1] 191898783962510625
LnUnrootedSplits(c(2,4))
#> [1] 2.70805
LnUnrootedSplits(3, 3)
#> [1] 2.197225
Log2UnrootedSplits(c(2,4))
#> [1] 3.906891
Log2UnrootedSplits(3, 3)
#> [1] 3.169925
NUnrootedSplits(c(2,4))
#> [1] 15
NUnrootedSplits(3, 3)
#> [1] 9