A link between the Matsumoto-Yor property and an independence property on trees

We prove that an independence property established by Matsumoto and Yor [2001. An analogue of Pitman's 2M-X theorem for exponential Wiener functional, Part II: the role of the generalized inverse Gaussian laws. Nagoya Math. J. 162, 65-86] and by Letac and Wesolowski [2000. An independence property for the product of GIG and gamma laws. Ann. Probab. 28, 1371-1383] is, in a particular case, a corollary of a result by Barndorff-Nielsen and Koudou [1998. Trees with random conductivities and the (reciprocal) inverse Gaussian distribution. Adv. Appl. Probab. 30, 409-424] where, for finite trees equipped with inverse Gaussian resistances, an exact distributional and independence result was established.