As a front-running disclaimer: I am not affiliated with the Sergeant Audie Murphy Club in any manner at this point in time.
Your question of the significance of the oak tree that is near his grave and that of so many other graves could indirectly be linked to him as well.
Original quote from http://warfoto.com/AudieMurphy.htm
" Audie Murphy was buried yesterday beneath a white oak tree in Arlington National Cemetery in a ceremony that briefly brought together Murphy's disparate worlds as most decorated soldier and undistinguished cowboy actor. "
"Oak leaves are traditionally an important part of German Army regalia. They also symbolize rank in the United States Armed Forces. A gold oak leaf indicates an O-4 (Major or Lt. Commander), whereas a silver oak leaf indicates an O-5 (Lt. Colonel or Commander). Arrangements of oak leaves, acorns and sprigs indicate different branches of the United States Navy Staff corps officers. Oak leaves are embroidered onto the covers (hats) worn by field grade officers and flag officers in the United States armed services.
If a service member earns multiple awards of the same medal, then instead of wearing a ribbon or medal for each award, he or she wears one metal representation of an "oak leaf cluster" attached to the appropriate ribbon for each subsequent award."
Original quote from http://www.fleurdelis.com/symbolism_LMNOP.htm#Oak
(a website on heraldry:
"Oak Tree, Leaves, Bush: Great age and strength; Heroism and victory"
Also from the above website on colors:http://www.fleurdelis.com/meanings.htm
White (Argent) Peace and sincerity
or if the tree really is a black oak:
Black (Sable) Constancy or grief
There is also a list of memorial trees within Arlington National Cemetery, not sure if that is one of them but here's a list of all. He's buried in section 46 and there are three oaks there (by my count from the list) that are memorial trees: http://www.arlingtoncemetery.m...C_Memorial_Trees.pdf
So perhaps the tree there does not mean anything for MAJ Audie Murphy's death but for heraldric purposes (depending on the color) the tree stands for Heroism and if the website above listed it correctly the 'White Oak' signifies peace and heroism. Major rank is signified by an Oak leaf (but the tree was there before he passed so this is just coincidence or extreme forward planning...) and if it IS one of the memorial trees well then it represents a living memory of the group it was planted/dedicated for.
Short of emailing Arlington to ask if it is one of the memorial trees or if it was planted for any specific reason that's as good an answer as I could find.