In　this　paper,　the　authors　firstly　construct　a　graph-theoretic　decomposition　of　an　index　set　for　tree　martingales,　and　based　on　this　decomposition,　they　give　a　locally　finite　tree　martingale＇s　notion　and　a　tree　martingale　decomposition　theorem.　Secondly,　they　establish　some　relations　between　the　locally　finite　tree　martingales　and　the　multiparameter　martingales,　and　furthermore　the　convergence　of　tree　martingales　is　shown　by　using　the　multiparameter　martingale　theory　of　Cairoli-Walsh.　Finally,　with　some　mild　conditions,　two　inequalities　for　tree　martingales　are　obtained　by　using　their　decomposition　theorem　and　multiparameter　martingale　theory.　(C)　2009　Elsevier　B.V.　All　rights　reserved.