It is unclear precisely what connection there was between Belecthor's death and that of the White Tree. It would be natural to take this as mere coincidence, but the text of The Lord of the Rings seems to imply some kind of causal connection: 'When Belecthor II, the twenty-first Steward, died, the White Tree died also...' (The Lord of the Rings Appendix A I (iv), The Stewards).
The idea of a tree failing as a great leader dies is not an uncommon one (for instance, according to Roman legend the deaths of the earlier Emperors were marked by the wilting of certain laurel trees). There is, however, no record that Belecthor II was a particularly outstanding leader in any field, so the implication of the loss of the White Tree is something of a mystery. It may be that Belecthor was in fact a great military hero, or a benefactor of his people, or even a specially selfish or villainous Steward (the Roman legend was tied most strongly to Nero's death, so this last possibility is at least plausible). History has forgotten the story of Belecthor's rule, however, and the full reason for the White Tree's death - if there was any reason at all - can never now be known.