if I put in a new lawn that is shared between my household and a tenant.
A "new lawn" (however you define just *exactly* what that is) is a property improvement. While land improvements do add to the cost basis, land is not a depreciated asset. Some "land improvements" are depreciated. But a "new lawn" doesn't sound to me like it is. If all you did was put down new sod, then while it can add to the cost basis, it's not depreciated and will have absolutely no impact on the depreciation you are required to take on rental property.
new lawn that is shared between my household and a tenant.
Sometimes, though not always, f you are renting out a physically separate structure on your property (such as an ADU) then the property improvement is either on the land specifically allocated to the tenant or land specifically allocated for personal use to you. But regardless of weather that applies to your specific and explicit situation or not, it's not going to make any difference to the depreciation you are allowed/required to take.
