Using　a　simple　function　of　the　sizes　of　two　trees　and　the　edit　distance　between　them,　this　paper　presents　a　normalized　tree　edit　distance,　which　not　only　satisfies　the　triangle　inequality,　but　also　is　a　genuine　metric　valued　in　[0,　1]　under　the　condition　that　the　weight　function　is　a　metric　over　the　set　of　elementary　edit　operations　with　all　costs　of　insertions/deletions　of　the　same　weight.　Moreover,　the　distance　can　be　directly　computed　through　tree　edit　distances　and　it　has　the　same　computing　time　complexity　with　the　tree　edit　distance.　Handwritten　digits　recognition　experiments　show　that　the　metric　normalization　can　reach　91.6%,　which　is　0.2%　and　0.8%　better　than　that　of　the　other　two　existing　methods　respectively　when　the　AESA　is　used.