The purpose of this paper are investigations on the structure of the reduced family tree of a positively regular and positively recurrent ([6, 101) subcritical GAITON-WATSON-Process (GWP) with a countable set of types, especially the generalization of the YAOLOM limit theorem and the source time theorem.The reduced family tree of a GWP with respect to t (we call it for abbreviation reduced t-family tree) is arising from the complete family tree of this GWP up to the tth generation under the condition that the tth generation is not empty in the following way: all and only those particles and the corresponding branches of the tree from the first up to the (t -l)th generation having no descendants in the tth generation are extincted.The YAGLOM and the source time theorem may be regarded as theorems on the (total) reduced family tree being in some sense the limit of the reduced t-family tree if t tends to infinity. The YAGLOM theorem says that the last generation in the reduced t-family tree tends in distribution for t 3 00 toward a random population having with probability 1 finitely many particles. The source time in the reduced t-family tree (we call it t-source time) is the random length of the top, i.e. the difference between t and the number of the last generation consisting of only one particle. If t tends to infinity the t-source time converges in distribution to a real random variable, which is denoted by the (total) source time. This proposition we call source time theorem.The source time theorem was originally proved using the YAOLOM theorem ([l, 31).
K. FLEISOHMANN and R. SIEOHUND-SCRULTZE recognized that the reduced t-familytree of a GWP is the family tree of a temporally inhomogeneous branching process, which we call the reduced t-branching process. Using this fact they proved for the single type case the source time theorem directly and derived from it the YAOLOM theorem ([2, 41). This converse way is more simple than the original one. A generalization to the multitype case was given in [7] and [ll].Here, for the countable cane both methods are combined. At first the existence of a YAGLOM limit distribution is showed under some uniformity condition by reducing to the method in the single type case. Using this result essential statementson the reduced family tree are derived allowing to show under another uniformity condition the YAOLOM theorem, the source time theorem and other theorems on the (total) reduced branching process, i.e. the limit (in some sense) of the reduced t-branching process for t --f 00.All uniformity conditions are fulfilled in the non-rapid case. The YAOLOM theorem was proved for continuous time branching processes under other restrictions in [a].
