I agree with Mike Davis. You don't want to use milled lumber. You need completely straight grain that runs the full length. Otherwise, when you need it most, it will break on you, hopefully without you being injured as a result. If you are really set on building a walking stick this way, go out into the forest and select a very straight young sugar maple or beech tree sapling that is just slightly larger than your final dimension and use it, removing the bark and trimming it down to size. If it dries straight, then shape the end and glue the ball to it. The down side is that this is going to be ONE VERY HEAVY WALKING STICK. Do you realize that "all trails are up hill"? At least it feels like this because that's the part of the trail that you will remember most.
I've hiked about 1/3 of the Appalachian Trail as well as many other trails in the Northeast, spending as much as 3 weeks at a time walking in the woods for distances of 100 - 180 trail miles at a time. When hiking, it won't take you very long to realize that the weight (actually the lack of weight) of everything that you take with you becomes very important. It won't take you very long to figure out what is really needed and what you can do without. I even went so far as to trim the borders off of my maps to reduce their weight. It really becomes that important. If you don't shed the weight as much as possible, you will not enjoy your hiking. In fact, in the first couple of miles on a trail you will frequently see items that were discarded and left behind by those ahead of you on their first hikes who decided that they just couldn't carry them any further.
My hiking stick has always been a piece of Bamboo, because it was the lightest, sturdy stick that I could find. Over the years it has acquired wraps of duct tape as it began to split length wise, but the splits don't significantly weaken it, as long as they are held together with the duct tape. It is still very strong, even though it's now about 60 years old. The top end handle area was also wrapped with black vinyl electrical tape about 40 years ago, to smooth it and prevent blisters. There's no ball on mine. It would be too heavy. My heart problems won't let me hike any longer, but I can't bring myself to throw away that bamboo walking stick. It's been a significant part of my life for a very long time.
Charley