The stairs *should match the "public" area flooring (as in the main living area floors). If the upstairs is for bedrooms and games room, this would be considered "private" (for the homeowners). If the main floor is the "public" area (living room, kitchen, dining, etc) then the stairs going UP should match the main floor flooring. The reason: the stairs are visible to the public and therefore they should match the public area floors. Now for tiles in the upstairs area. This is not common but it has been done. It takes extra work (and extra materials = extra money) to get this to work over wood-joist construction. The subfloors will have to be assessed for deflection rating. I'll assume they rating isn't good enough for tile (tile is VERY HEAVY and requires very STIFF/thick subfloor). That means you'll need another layer of plywood over top. That plywood will change the height of the subfloor at the TOP of the stairs...which can mess up the stairs themselves (according to code). And then you have to add the tile which can often add as much as 3/4" height - which can again mess up the top step which messes up the stairs yet again. And then we get into the "tile is loud" concept. It is extremely loud. It transfers noise "up and down" (from upstairs to down stairs) as well as "wall to wall" (down hallways, into the bedrooms, etc). To deal with the noise and echo, you could be looking at changing the doors to solid core (expensive proposition depending on how many doors you have). And if you find the noise transfer TOO much (up to down) then you may have to rework the ceiling on the main floor (pull it down, stuff it with acoustic insulation, replace ceiling with acoustic drywall, etc). You can see the $$$$ signs flying out the window. Please look at the upstairs as a part of the whole home. Ask yourself how big your MAXIMUM budget is (once all the bills are paid = includes labor as well). And then find out if it can handle the "extras" such as new subfloor and possibly a stair case that has to be updated to allow for the extra height at the top of the stairs (rebuilding stairs is one of the most expensive projects in the building industry). Then make your decision as to whether or not tile on the second floor is a good idea FOR YOU. You are the only one who can answer that question.

Have your stair treads refinished to match whatever stain you pick for the wood floors.

Your stair treads appear to be stained. Simply sand them and then refinish them in a natural finish. Be aware that the stair treads MAY BE red oak (which is CHEAPER than white oak). If they are white oak, a natural clear coat (water based polyurethane) will NOT amber or turn a deep orange. I think this has happened to the stairs...or they have a medium stain added. It is hard to tell from your photos. Please add a close up of a stair tread. Please try to photograph a step with the bamboo showing as well as a tread with the upstairs oak showing. That way we can see what is happening. Normally you try to match the stair tread with the PUBLIC area flooring - which I'm assuming is the bamboo. The private areas (normally the upstairs where the bedrooms exist) are stand-alone areas and do not need a match to the stairs.

Red oak and white oak are both "oak" . Which is the same as saying they are both TREES. But so too are cherry, maple , walnut. fir, African Mahogany and lowly pine! in NO WAY do they take stain color the same way. A factory finish won't get the red out of a red oak, and nor will an on site finisher unless it is water popped to go quite dark. Grain can appear nearly same or quite different, depending how it was cut. "Despite their similar names, these two woods are very different. there is only one true species of red oak (Quercas rubra) and of white oak (Quercas alba), but there are 20 other varieties commonly grouped under the “red oak” or “white oak” heading based on their characteristics, so the possible color variations can be significant." You missed Oak and Broads, point. PICK ONE floor and match the treads.. Rather than make brain surgery of a relatively simple decision? : )