Will I need a converter for my USA laptop in Botswana?

Since 230V is in the range 100-240V, you do not need a transformer. You will probably need an adapter. Botswana doesn't seem to have a standard; can you find out what kind of plugs are in your hotel, or the business you're visiting?

I would suggest bringing at least one adapter labeled Africa and a British one, but consider a universal adapter so you can work with whatever you find:

My universal didn't work in South Africa; the hotel, however, sold adapters. You may end up buying one there once you know what plug you are working with.

Here's a canonical reference for Peter Green's answer (sorry, too long for a comment).

Botswana uses Type D, Type G and Type M sockets. See:

• http://www.iec.ch/worldplugs/typeD.htm
• http://www.iec.ch/worldplugs/typeG.htm
• http://www.iec.ch/worldplugs/typeM.htm

noting the panel to the right contains 'Botswana' in each case.

These are specified here: https://en.wikipedia.org/wiki/AC_power_plugs_and_sockets:_British_and_related_types

Type G are BS1363 square-pin British sockets (i.e. what you now see in the UK). The plugs look like this:

(source: Wikimedia commons)

And the sockets look like this:

(source: Wikimedia commons)

Types D (5A) and M (15A) are the old round-pin British 5A and 15A sockets (BS546) which are hardly ever seen in the UK now save for some specialist applications where fused plugs are undesirable (e.g. stage lighting where changing fuses is difficult). They are much better known as South African standard (SABS1661 is the SA standard for the 15A variant, which is allegedly more common for power outlets), which is why they are used in Botswana.

The plugs look like this (left hand 15A, middle 5A, right hand an irrelevant 2A plug):

(source: Wikimedia commons)

and the sockets look like this (left hand 15A, right hand 5A):

(source: Wikimedia commons)

These are relatively unusual on the 'plug' side of universal adaptors (i.e. adaptors that support multiple types of power sockets, as well as allowing multiple things to be plugged into them). You can however buy adaptors that plug into only these sockets, and support multiple types of plug - e.g. (for example) here (unaffiliated link) and here (Amazon, unaffiliated link).

You do not need a transformer as (per Kate Gregory's answer) your laptop power supply will take between 100V and 240V at either 50 or 60Hz, and the supply is 230V (within the voltage range) at 50Hz (one of the options). You merely need a physical adapter (as per the above). Most modern power supply units are similar, but it is best to check (as you did).

You don't need a transformer, but you will need one or more plug adaptors. From some googling it seems that Botswana uses a mixture of British 5A, 13A and 15A plug types.

Unfortunately in my googling I failed to find a source of adaptors for connecting American plugs to 5A and 15A BS546 sockets that I was happy to recommend. The only ones I found had horrible universal sockets (the type that fits lots of plugs, none of them very well) and the BS1363 adaptor in the range was unfused, which doesn't inspire confidence (BS1363 plugs should always be fused).