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