Geodesic
Occupation times for countable Markov chains~III. Chains on a~tree with one branching point
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IX, Tome 142 (1985), pp. 25-38

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There is given a detailed description of the Green function for Markov chains on a tree with one branching point. In the case of chains with continuous time, one proves the Markov property of the field of the occupation times. 
@article{ZNSL_1985_142_a1,
     author = {S. S. Vallander},
     title = {Occupation times for countable {Markov} {chains~III.} {Chains} on a~tree with one branching point},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {25--38},
     publisher = {mathdoc},
     volume = {142},
     year = {1985},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1985_142_a1/}
}
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S. S. Vallander. Occupation times for countable Markov chains~III. Chains on a~tree with one branching point. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IX, Tome 142 (1985), pp. 25-38. http://geodesic.mathdoc.fr/item/ZNSL_1985_142_a1/