Here's a set of six fully authentic arrows I'm working on for a customer in Canada. The shafts are lime, and are tapered with a torpedo profile - full diameter for most of the shaft to keep the weight and stiffness up, and then dropping suddenly towards the nock.
The nock end is around 11/32 as seen on many of the Mary Rose arrows, and the horn insert is fitted using a sharp knife, running down the nock and splitting it open instead of sawing the slot. This results in a very thin, very tight slot which is just perfect for the carefully prepared cow horn inserts themselves. The horn is pared to thickness with a knife, which leaves a tapered point and thicker end section, and as this is pushed into the slot a lovely wedge-shaped horn insert is the end result. Again, this can all be seen on the original Mary Rose arrows and is a big difference to what is usually displayed as "medieval" today - thick black buffalo horn inserts certainly weren't used back then!
The lovely thing about cow horn inserts is that at the right angle you get a brilliant honey glow as the blonde translucent horn picks up the light.