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We develop an on-line wavelength assignment (WA) algorithm for a wavelength-routed WDM tree network. The algorithm dynamically supports all -port traffic matrices among end nodes, where denotes an integer vector and end node , can transmit at most wavelengths and receive at most wavelengths. Our algorithm is rearrangeably nonblocking, uses the minimum number of wavelengths, and requires at most lightpath rearrangements per new session request, where is the degree of the most heavily used node. We observe that the number of lightpath rearrangements per new session request does not increase as the amount of traffic scales up by an integer factor. In addition, wavelength converters cannot reduce the number of wavelengths required to support -port traffic in a tree network. We show how to implement our WA algorithm using a hybrid wavelength-routed/broadcast tree with only one switching node connecting several passive broadcast subtrees. Finally, using roughly twice the minimum number of wavelengths for a rearrangeably nonblocking WA algorithm, we can modify the WA algorithm to be strict-sense nonblocking.