determining a lower bound price of a group of {TAIL ~ bids} of a subtree of {HEAD ~ the search tree} 
determining a lower bound price of a group of { TAIL ~ bids } of a subtree of { HEAD ~ the search tree }
OTH OTH OTH OTH OTH OTH OTH OTH OTH TAIL TAIL TAIL TAIL TAIL OTH OTH OTH OTH HEAD HEAD HEAD HEAD HEAD HEAD HEAD