takes 6 (non-negative) distances as arguments (in the order [d12, d13, d14, d23, d24, d34]) and returns tree (with branch lengths if the four-point condition is met).

buneman_four_point_condition_tree.py

#!/usr/bin/env python
from __future__ import print_function
import sys
try:
    farg = [float(i) for i in sys.argv[1:]]
    for i in farg:
        assert i >= 0.0
    d_12, d_13, d_14, d_23, d_24, d_34 = farg
except:
    m = 'd_12, d_13, d_14, d_23, d_24, d_34'
    sys.exit('Expecting 6 (non-negative) distances in the order:\n {}\n'.format(m))

bs_12 = d_12 + d_34
bs_13 = d_13 + d_24
bs_14 = d_14 + d_23
pref_s = min(bs_12, bs_13, bs_14)
if bs_12 == pref_s:
    tree = '(1, 2, (3, 4))'
    sep = min(bs_13, bs_14) - bs_12
    exp_0 = abs(bs_13 - bs_14)
elif bs_13 == pref_s:
    tree = '(1, 3, (2, 4))'
    sep = min(bs_12, bs_14) - bs_13
    exp_0 = abs(bs_12 - bs_14)
else:
    assert bs_14 == pref_s
    tree = '(1, 4, (2, 3))'
    sep = min(bs_12, bs_13) - bs_14
    exp_0 = abs(bs_12 - bs_13)

print("The preferred tree is: {}".format(tree))
print("It is preferred by a margin of {}".format(sep))
print("The difference in the 2 larged sums is expected to be 0\nIt was calculated to be {}".format(exp_0))

ROUND_TOL = 1.0e-8
if exp_0 <= ROUND_TOL:
    internal = sep / 2.0
    if bs_12 == pref_s:
        treet = '(1:{fir}, 2:{sec}, (3:{thi}, 4:{fou}):{internal});'
        diff_fir_sec = d_13 - d_23
        sec = (d_12 - diff_fir_sec) / 2.0
        fir = d_12 - sec
        diff_thi_fou = d_13 - d_14
        fou = (d_34 - diff_thi_fou) / 2.0
        thi = d_34 - fou
    elif bs_13 == pref_s:
        treet = '(1:{fir}, 3:{thi}, (2:{sec}, 4:{fou}):{internal});'
        diff_fir_thi = d_12 - d_23
        thi = (d_13 - diff_fir_thi) / 2.0
        fir = d_13 - thi
        diff_sec_fou = d_12 - d_14
        fou = (d_24 - diff_sec_fou) / 2.0
        sec = d_24 - fou
    else:
        treet = '(1:{fir}, 4:{fou}, (2:{sec}, 3:{thi}):{internal});'
        diff_fir_fou = d_12 - d_24
        fou = (d_14 - diff_fir_fou) / 2.0
        fir = d_14 - fou
        diff_sec_thi = d_12 - d_13
        thi = (d_23 - diff_sec_thi) / 2.0
        sec = d_23 - thi
    
    template = "With branch lengths:\n {}".format(treet)
    print(template.format(fir=fir, sec=sec, internal=internal, thi=thi, fou=fou))