Can my tree stood in water after the yard flooded and now it looks sick be fixed without removing the tree?

Depends on species and duration more than on anything you do. Silver maple, sycamore, swamp white and pin oak, river birch, baldcypress, and willow tolerate extended growing-season flooding; sugar maple, most conifers other than baldcypress, yew, dogwood, cherry, and beech are damaged quickly. What helps: get the water off — clear the blockage, redirect the downspout, cut a drainage path — then stay off the soil entirely while it is wet, because foot and vehicle traffic on saturated soil compacts it worse than the flood did. Mulch, skip fertilizer, and water through the drought that often follows, since a damaged root system cannot take up water even when it is there. Do not add soil to level the low spot; that buries the root collar and is the most common way people finish off a tree they were trying to rescue. Removal becomes the call when the root collar is decayed, conks appear at the base, or the tree has shifted.