The first, 50, terms of the sequence of (truncated) average of the r.v. "tota\
    l height" defined on rooted labeled trees is



[0, 1, 2, 4, 7, 10, 14, 17, 22, 26, 31, 36, 41, 47, 53, 59, 65, 72, 78, 85, 93,
100, 108, 115, 123, 131, 140, 148, 157, 166, 175, 184, 194, 203, 213, 223, 233,
243, 253, 264, 275, 285, 296, 307, 319, 330, 341, 353, 365, 377]


The first, 50, terms of the sequence of the modes (most likely total height) \
    of the r.v. "total height" defined on rooted labeled trees is



[1, 1, 3, 4, 7, 9, 13, 18, 20, 25, 29, 32, 38, 43, 47, 54, 60, 65, 71, 78, 84,
91, 97, 104, 112, 119, 127, 134, 142, 150, 158, 166, 175, 184, 192, 201, 210, 
220, 229, 238, 248, 258, 267, 277, 288, 298, 308, 319, 329, 340]


The first, 50, terms of the sequence of the (truncated) medians of the r.v. "\
    total height" defined on rooted labeled trees is



[0, 1, 3, 5, 7, 11, 14, 18, 22, 26, 31, 36, 41, 46, 52, 58, 64, 70, 77, 84, 91,
98, 105, 113, 120, 128, 136, 145, 153, 162, 171, 180, 189, 198, 207, 217, 227,
237, 247, 257, 267, 278, 288, 299, 310, 321, 332, 343, 355, 366]


    Finally, the actual most likely total-heights for i from 1 to, 50,  are



[0, 2, 6, 24, 140, 1470, 14700, 212016, 3344544, 65651040, 1375189200, 
34656763680, 927629033760, 28171817834160, 910348005326760, 32513978827293600,
1231904860602736320, 50514953476689374400, 2192716816985859877440, 
101601763687062300758400, 4969502345083890644009280, 
257073838926916292295814800, 13973334519698516240097518160, 
798471846115888194812553425280, 47760230138758807448275103376000, 
2987742536120434428388248067872000, 194918958352179663149623163975172000, 
13249918410915772280941893051659654400, 936467137474908934827878402882646312000
, 68725096462489664047136970882391541712000, 
5230989289713737175362211759761145932172000, 
412334890919895340685040307512061602635504640, 
33623469469203194907634438632979476697382657280, 
2832888372551898382125606330654219137385434465280, 
246376884039611569938442644686381729393336401689600, 
22096890749443827672457265621372776660764565001740800, 
2041824202024076467961635467252357405585059108083046400, 
194211409021974514331022462843322023209423619765914096640, 
19001105278787459840870413787562083643657161505093396372960, 
1910548601195347712951623206611633321328411100611078362841600, 
197297453876297770615997977001667748628867281970678770437062400, 
20909812812628797049368911393852435325770927879531322715163769600, 
2272887234045178833641640547744232366661773478413281264892119488000, 
253237310827710829063209421110800830749506148249985544559140824179840, 
28902649229997733595213783426600767661414475560902700194026347823763200, 
3377319937364633180500100297007226307143394517069540311680168507772640000, 
403818004247051388096899095613624243531346608964309954297307352864023353600, 
49379767327049458170679402777034477581177338024756181640717693737634954188800,
6172604484659077540565379052129527846338250059105244340711486762166523746353920
, 78835370453832259840014014041005616601511198925818597134255673690941103678220\
8000]